Falkinger proposes incentive mechanism which relies on a simple rule of payoff and penalty depending on the difference between agents contribution and a mean contribution\cite[p.250]{szkopy00}. Mean donation is calculated among agents of akin incomes. In order to carry it out the set of agents is being divided into a number of income classes. Agent's payoff is figured as follows:
\begin{equation}
r_i=\beta (x_i-\frac{x^m_{-i}}{n_m-1})
\end{equation}
Where $n_m$ is the number of members of the $m$th income class and $x^m_{-i}=\displaystyle\sum_{j\in I_m-\{i\}}x_j$. $I_m$ is the $m$th income class and $\beta\in[0;1]$ is a subsidy rate. 
When $r_i<0$ the payoff becomes a penalty to be paid.
The budget constraint becomes:
\begin{equation}
\begin{split}
c_i+x_i&=y_i+r_i \\
&\Downarrow \\ 
c_i+x_i&=y_i+\beta (x_i-\frac{x^m_{-i}}{n_m-1})
\end{split}
\end{equation}
while the donation turns into:
\begin{equation}
P_g\cdot G = \sum_{i=1}^n (x_i - r_i)= x_i - r_i + x_{-i} - r_{-i}
\end{equation}
but if we analyze it carefully, we will find out that whatever value $\beta$ has, penalty/reward rule redistributes somehow the sum of $x_i$'s among all agents so $P_g$ remains the same. The only change it makes is the incentive of paying more in order to make the mean donation higher.

%Let's examine the following example:
%
%To the income class $m$ belong two agents: $i$ and $-i$. Their incomes are given. They decide to donate $x_i$ and $x_{-i}$. Thus $P_g\cdot G = x_i + x_{-i}$ but the real donations would be respectively $x_i-r_i$ and $x_{-i}-r_{-i}$. So, having $n_m=2$:
%
%\begin{equation}
%x_i - r_i = x_i - \beta (x_i-x_{-i}) = (1-\beta)x_i + \beta x_{-i}
%\end{equation}
%and
%\begin{equation}
%x_{-i} - r_{-i} = x_{-i} - \beta (x_{-i}-x_i) = (1-\beta)x_{-i} + \beta x_i
%\end{equation}
%
%Therefore the private consumption reaches:
%
%\begin{equation}
%c_i = y_i - (1-\beta)x_i - \beta x_{-i}
%\end{equation}
%and
%\begin{equation}
%c_{-i} = y_{-i} - (1-\beta)x_{-i} - \beta x_i
%\end{equation}
%
%As we can see, $\beta=\frac{1}{2}$ equalizes the final donation among agents within income class. %$\beta > \frac{1}{2}$ forces agents who wanted to pay less than average
